Natural frequencies of an oscillator attached to a two-segment string are determined using the algebraic perturbation theory. It is found that the characteristic equation possesses two structurally different classes of the natural frequencies. The necessary equation describing the motion of the oscillator is constructed and is shown that when the difference in the mass densities of the segments is negligibly small, the imaginary parts of the frequencies belonging to the first class diverge logarithmically, leading to a vanishing contribution due to the quickly damped oscillations. So, in the uniform string limit the motion of oscillator is governed by the second class of complex frequencies. Based on this observation, a connection between the present analysis with previous works is established.
